Yamada and Knight (2001) presents a generative tree-based model that is trained using the EM algorithm, thus aligning the words in the parallel corpus while extracting syntactic transfer rules. Syntax trees are provided by automatically parsing the English side of the corpus in a pre-processing step. They also present a chart parsing algorithm for their model

Kenji Yamada and Kevin Knight (2002): A Decoder for Syntax-Based Statistical MT, Proceedings of the 40th Annual Meeting of the Association of Computational Linguistics (ACL)

Relaxing the isomorphism between input and output trees leads to the idea of quasi-synchronous grammars (QG), which have shown to produce better word alignment quality than IBM models, but not symmetrized IBM models